Analysis of the harpsichord plectrum-string interaction

D. Chadefaux, J-L. Le CarrouLAM / d’Alembert,UMR CNRS 7190,

UPMC Univ Paris 06

S. Le ConteLaboratoire de Recherche

et Restaurationdu Musee de la Musique

M. CastellengoLAM / d’Alembert,UMR CNRS 7190,

UPMC Univ Paris 06

ABSTRACT

This paper describes a study of string plucking for the harp-sichord. Its aim is to provide an experimentally-based anal-ysis of the plectrum-string interaction and to propose somerefinements of an existing model. An experimental setuphas been designed using a high-speed camera combinedwith a laser doppler vibrometer and classical audio record-ings. This provides accurate estimations of jack and plec-trum motion throughout the harpsichord plucking in a re-alistic musical context. The set of descriptors extractedfrom these measurements provides typical orders of mag-nitude of plucking parameters required to feed and vali-date the investigated model. Results highlight estimationsof the instrumentalist’s control parameters as well as of theintrinsic plucking parameters according to the performedsequence tempo. Besides, a model of the plectrum-stringinteraction, which takes into account the section variationat the plectrum tip, gives results close to the experimentalones. This model will be of great interest to enquire aboutharpsichord plectrum voicing.

1. INTRODUCTION

Among all elements of a harpsichord, the voicing process,which directly deals with the excitatory mechanism tun-ing, has to be investigated. The harpsichord plucking ac-tion mostly consists in a piece of wood moving vertically,called the jack, while the instrumentalist’s finger is de-pressing the key. It conveys the plectrum attached to thejack to pluck the string. Although traditionally made ofquill [1], the plectrum is nowadays usually made of delrin.During the voicing process, the maker tunes the plectra toobtain the best equilibrium among the different choirs (8and 4ft), the bass and the trebles. Practically, it impliesto carve each plectrum to adjust the plectrum-string inter-action according to the instrument’s sound and the instru-mentalist’s touch. Several kinds of voicing can be outlinedin the Musee de la musique’s collection; the three mainones are: hard, soft and medium.

The physics of harpsichord has already been investigatedin several studies, focusing on its elements [1,2] as well ason its modal behavior [3, 4] or on its dynamics [5]. Fur-ther, the question of the plucking action itself has mostly

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been addressed through modeling [6–8]. Besides, the influ-ence of plectrum geometrical parameters on the producedsound has been theoretically [9] and experimentally [10]analyzed for the harpsichord and the classical guitar, re-spectively. Although few studies have enhanced the harp-sichord plucking modeling by measurements [11, 12], noinvestigation in a musical context has been carried out asfor instance for the concert harp [13, 14].

The present paper describes preliminary results from anongoing exploratory study of the plectrum voicing. We in-vestigate harpsichord plucking action, within the plectrum-string interaction analysis and modeling. For this purpose,we first present the experiment carried out in order to mea-sure plectrum and string motions during playing. Then, weprovide a experimentally-based description of the pluckingaction. Finally, a plectrum-string interaction model is con-fronted to experimental data.

2. EXPERIMENTAL PROCEDURE

2.1 Experimental protocole

A set of measurements is fulfilled in order to investigatethe harpsichord plucking action. The experimental proce-dure illustrated in Fig. 1 consists in capturing the plectrum-string interaction during a harpsichord performance. Asobservations indicate that no motion occurs along the stringaxis (denoted ~ez in Fig. 2), the analysis is performed in the(~ex, ~ey) plane. For this purpose, plectrum motion is mea-sured through a high-speed camera set at 10000 frames persecond focusing on the plectrum. A laser doppler vibrom-eter is also focusing on the associated key to measure itsvelocity within the performance. The laser beam is fo-cused at about 3 cm from the tip of the key, avoiding ob-struction from the musician hand while depressing the key.Simultaneously, acoustical signals are recorded with a mi-crophone, allowing the synchronization of the database.

A skilled harpsichord player has been asked to performthe sequence presented in Fig. 3 with two various tempiwhich have been estimated at about 230 bpm and 440 bpm.Besides, as the high-speed camera sampling rate induces arestricted resolution of 208px×200px representing a12.3mm×11.8mm area, we were able to only investigateone note. Based on the harpsichord player advices, G4,which fundamental frequency is 392 Hz, is chosen. Thisprocess has been carried out with a plectrum in delrin shownin Fig. 4.

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Figure 1. Experimental setup.

l

Figure 2. Scheme of the plectrum-string interaction anddefinition of the parameters.

2.2 Data processing

Fig. 5 shows an example of image obtained through thehigh-speed camera. The entire sequence is processed todetermine a set of plectrum-string interaction descriptorsas follow. The desktop background defined at the plectrumrest position is first subtracted from all images. Then, thearea of interest containing the plectrum is selected by theuser. This template is recursively searched in the entiresequence through a block-matching algorithm model [15].The string cannot be distinguished in the image becauseof its contrast. The latter is then refined to obtain a blackand white image where only the plectrum appears in white.Hence, based on the framing projection relatively to themeasurement plane (~ex, ~ey , see Fig. 2), the trajectory ofthe string’s plucking point (xs, ys) in the latter plane canbe extracted from the sequence through its shadow on theplectrum.

Further, the synchronization of acoustical and key veloc-ity signals is needed to point out the instrumentalist con-trol. For this purpose, the onset of each sound event of

Figure 3. Score of the performed sequence.

Figure 4. Investigated plectrum: top and side views.

Figure 5. Example of image obtained through the high-speed camera.

the score presented in Fig. 3 is highlighted in the acous-tical signal, through a standard onset detection algorithm[16, 17]. These instants indicate the strings’ release by theplectrum.

3. HARPSICHORD PLUCKING DESCRIPTION

3.1 Temporal phases

The plectrum is directly governed by the instrumentalistto interact with the string through the harpsichord mech-anism. Thus, its motion reveals a part of the musician’scontrol while playing as well as determines the initial con-dition of the string vibration. The laser doppler vibrome-ter focused on the pressed key directly conveys the veloc-ity pattern of the plectrum at its connection point with thetongue. This investigation combined with the observationof the plectrum motion through the high-speed camera leadto a description into four temporal phases. Based on a harpplucking analysis [13], it can be decomposed as follow:

• The preparatory movement: the plectrum raises fromits rest position and approaches the string, ∀t ∈ [ti, tc].Usually, the player first pushes the key until reachingthe string to feel its contact with the plectrum. Theplectrum is then kept just below the string.

• The sticking phase: the plectrum moves the stringin the vertical direction from its rest position, ∀t ∈[tc, ts];

• The slipping phase: the string slips on the plectrumsurface towards its free end, ∀t ∈ [ts, tr];

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• The free oscillations of the string, ∀t > tr.

The determination of each instant tc, ts and tr is basedon high-speed camera films. The sticking instant tc is au-tomatically detected at the first instant where the string ismoved up by the plectrum. Then, the beginning of the slip-ping phase ts is defined when the string’s displacementgets a horizontal component. Finally, the release instantis obvious to determine since it corresponds to the begin-ning of the plectrum vibrations. Note that tr is manuallyestimated. Although the detection of tc and tr are straight-forward, ts is more complicated to highlight.

In the present case, we estimate the entire duration of theharpsichord plucking action at 12± 2 ms and at 7.9± 0.6 msfor the normal and the fast tempi, respectively. The mea-surements reported in the literature are about 80 ms [11]and 25 ms [12]. Differences between these orders of mag-nitude can be explained by the various musical contextsconsidered. Further, the slipping phase duration has notbeen pointed out in these previous studies. However, basedon the plucking analysis, we measure that the slipping phaselasts about 36 ± 3 % and 26 ± 4 % of the entire plectrum-string interaction duration for the normal and the fast tempi,respectively. The small uncertainties, computed over eachplayed note with a 95 % confidence interval, tend to in-dicate that the measurement protocol is reliable. Eventu-ally, remark that the first note has not been taken into ac-count since it corresponds to the instrumentalist adaptationto the plectrum, which implies a longer interaction: 26 ms,i.e. about two times longer than the following pluckingactions.

3.2 Jack velocity

The jack motion is directly measured through the high-speed camera during the plectrum-string interaction. Thelinear regression associated to its vertical displacement in-dicates that the jack velocity is constant during plucking.Indeed, the computed coefficients of determination are0.97 ± 0.02, and 0.99 ± 0.00 for the normal and the fasttempi, respectively. Eventually, the jack velocities are es-timated in the two former contexts at 0.10 ± 0.05m/s and0.25 ± 0.03m/s. These estimations are relevant regardingthe tempo of the played sequence: the higher the tempo,the higher the jack velocity. Besides, it confirms that theinstrumentalist control is repeatable along a musical se-quence, as expected for an expert gesture [18]. Finally,these measurements are consistent with those previouslymeasured or used to fed a harpsichord plucking model [8,11]. However, note that these values are highly dependenton the plectrum’s shape and the material as well as on thestring’s properties.

Because of the arduousness of the high-speed camera filmspost-processing, the laser doppler vibrometer is used to getan insight on the instrumentalist’s control during the entiresequence. Fig. 6 shows the G-key velocity curve measuredon one plucking action in a musical context. This curve hasbeen chosen because of its representativeness among thewhole database. It indicates that the player depresses firstthe key until the plectrum reaches the string ∀t ∈ [ti; tc].

Figure 6. G-key velocity curve during a plucking actionand its preparation. ti corresponds to the instant where thefinger starts pushing the key. The plectrum-string interac-tion begins at tc, while tr and tf are the instants where thestring and the key are released, respectively.

Then, during the plucking action (∀t ∈ [tc; tr]), the plec-trum velocity presents a second shape. Eventually, fromthe instant the string is released until tf , the plectrum ve-locity presents the same pattern than during the first phase.After this last instant, the finger releases the key toward itsrest position.

3.3 Initial condition of the string vibration

Although the free oscillations of the string do obviouslynot convey all the informations relative to the radiated sound,they reveal variations between plucking actions. Indeed,assuming the string flexible, stretched to a tension T, ofuniform linear density ρl and fixed at its end, the trans-verse vibration of the string along the vertical direction canbe described by [19, 20]

~r(z, t) =∞∑n=1

(An cos(2πfnt+ Ψn) (1)

+ Bn sin(2πfnt+ Ψn)) ~Φne−αnt,

where fn = nc/2L are the eigenfrequencies, Φn are themodal deflections, αn are the damping coefficients andΨn = arctanαn/ωn. The modal amplitudes An and Bnare written as

An =2Dtr sin(knz0)

k2nz0(L− z0), (2)

Bn =2Vtr sin(knz0)

k3nz0(L− z0)c, (3)

with kn the wavenumber, and z0 the plucking point alongthe length of the string. Dtr =

√xs(tr)2 + ys(tr)2 and

Vtr are the initial position and velocity of the string re-garding its rest position (see Fig. 2), respectively. Further,the magnitude of each polarization of the string is of greatimportance relatively to the radiated sound. It can be esti-mated by using the initial angle between the two compo-nents xs(z0, t) and ys(z0, t) of the transverse string motionduring the first vibration instants. It is referred to as Γtr ,as shown in Fig. 2.

Tab. 1 presents the initial conditions of the string’s vi-brations for the two performed sequences. Let us remind

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Slow Fastxs(tr) (mm) 0.27 ± 0.02 0.28 ± 0.02ys(tr) (mm) 0.71 ± 0.07 0.90 ± 0.01Vtr (m/s) 0.06 ± 0.02 0.15 ± 0.01Γtr (deg) 17 ± 2 20 ± 1

Table 1. Initial conditions of the string vibration. Themean is computed on eight samples. The reported uncer-tainty represents a 95% confidence interval.

that the jack velocity was substantially increased for thefaster sequence. As expected, the tempo does not influencethe string position in the horizontal direction: this value iscompletely determined by the initial position of the stringrelatively to the jack. However, results indicate that themore the jack velocity, the higher the string is released. Inthe present case, we measure an elevation of the string ofabout 27 % in the case of the faster tempo relatively to theslower one. This obviously implies a higher string’s veloc-ity magnitude as well as a more important force applied bythe string on the plectrum’s end. The latter directly con-veys to a higher plectrum’s deformation and therefore to amore important initial polarization angle Γtr .

4. COMPARISON BETWEEN MEASUREMENTSAND MODELING

In this section, a comparison between a mechanical modelof the plectrum-string interaction and experimental resultsis presented.

4.1 Plectrum-string interaction model

The model of the plectrum-string interaction presented inthis section is based on that of Perng and colleagues [7,8, 12]. In this model, based on the assumption that nofriction occurs between the string and the plectrum, thestring is seen as a punctual force applied perpendicularlyto the plectrum. The latter is considered as a cantileverbeam with small strain, made in an isotropic material, andwithout any twisting motion. Another assumption is theuniformity of the plectrum section. However, as shown inFig. 4, the plectrum is beveled under its tip to slip on thestring when the jack falls down. This thickness variationinduces a modification of the moment of inertia at the tip ofthe plectrum. In order to take it into account in the model,the plectrum is considered as a beam composed of two seg-ments: the first one from 0 to L1 with a moment of inertiaof I and the second one from L1 to L with a moment of in-ertia of I/8. With these assumptions, the deflection angleof the plectrum is

l ∈ [0− L1],Γ(l) =F

EI

(Ll − l2

2

), (4)

l ∈]L1 − L],Γ(l) =F

EI

(7

2L21 − 7LL1 + 8Ll − 4l2

),

where E is the Young modulus of the plectrum materialand F the force applied by the string at a distance l to theplectrum (see Fig. 2). The second Newton’s law is applied

String length Ls 47.5cmString tension T 85.7NPlectrum Young Modulus E 1.5GPaPlectrum length L 3.5mmBeginning of the bevel L1 2.9mmMoment of inertia I 0.011mm4

Table 2. Parameters of the plectrum and of the string.

Beam composed of one segment

Beam composed of two segments

Measurement data

Figure 7. Experimental and theoretical plectrum deflec-tion at the release instant. Referred axes are the same as inFig. 2.

on the string’s element in contact with the plectrum duringthe interaction. Its projection along x-axis and y-axis arewritten as

∆m∂2xs∂t2

= −Kxs + Fpx(l), (5)

∆m∂2ys∂t2

= −Kys + Fpy(l). (6)

With a geometrical approach,K is computed from the string’stension T , length Ls and the plucking ratio β using K =T/(LSβ(1− β)). Eqs. 5 and 6 indicate that the plectrum-string interaction depends on the string’s position on theplectrum l. This position can be extracted from the jackdisplacement (xj ,yj) by computing at each time step thefollowing system of equations (with the small angle ap-proximation):

xs(t)− xj(t) =

∫ l

0

1− 1

2Γ(x)2dx, (7)

ys(t)− yj(t) = −∫ l

0

Γ(x)− 1

6Γ(x)3dx. (8)

In this way, at each time, the string position (xs,ys), theplectrum deflection Γ and the plectrum force Fp may beknown.

4.2 Results

In order to compare experimental data to theoretical ones,the model is computed with parameters given in Tab. 2.

In Figure 7, the plectrum deflection at the release instantis shown. Unfortunately, all the plectrum’s shape cannotbe extracted because, at this instant, the string hides theplectrum tip. Nevertheless, theoretical results are found tobe consistent with the experimental data. In this figure,we also present the model based on a beam composed of

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one segment of moment of inertia I with the one with twosegments (as explained in Sec. 4.1). The proposed refine-ment induces a very slight modification of the plectrumshape prediction. However, the comparison of the plec-trum deflection angle (Γ(L) = Γtr ) for the two configura-tions indicate that for an unique segment the angle is 13.4◦

whereas for two segments it is 20.9◦. The second resultis closer to the experimental value (see Fig. 1). The beamcomposed of two segments is thus valuable to model theharpsichord plectrum.

Based on the plectrum-string model, the initial conditionsof the string vibration can be computed. At the release in-stant, we obtain xs(tr) = 0.31 mm and ys(tr) = 0.80 mm.These two values are very close to the experimental ones,especially for the fast case (see Tab. 1). Note that ourresults are obviously independent of the jack velocity be-cause no friction force is taken into account in the model.Therefore, when the string slips fast on the plectrum, lessfriction forces occur and the model is found to be moreappropriate than for the slow case.

5. CONCLUSION

This paper has presented an investigation of harpsichordplucking action. A well-controlled measurement setup con-veying the jack and plectrum motion features as well as theproduced sound has been carried out in a musical context.

As expected, results indicate that the plectrum-string in-teraction is longer when the playing tempo is slower. Thisis directly related to the jack velocity during the interac-tion. Indeed, a slower tempo leads to a lower jack veloc-ity while it is in contact with the string. Besides, it hasbeen shown that the plectrum velocity is constant duringthe plectrum-string interaction. Although these two param-eters can be considered as control parameters, the instru-mentalist cannot directly act on the initial conditions of thestring vibration, which highly contribute to define the pro-duced sound. However, a relationship has been pointed outbetween the playing tempo and the initial position, veloc-ity and initial angle of the string’s free oscillations. Indeed,we observed that the string is released higher relatively toits rest position for a higher playing velocity, which ob-viously imply a higher initial string’s velocity. Moreover,the string applies therefore a more important force on theplectrum, conveying to a higher plectrum bending and thenvariation in the initial string’s oscillations polarization. Adeeper investigation on several plectrum’s shape and mate-rial is required to conclude about the implications of theseobservations on the produced sound. Further, a plectrum-string interaction model has been implemented and con-fronted to our measurements. Simulations results indicatethat the variation of the plectrum geometry along its lengthhas to be taken into account in order to well-reproduce itsdeformations throughout the plucking action. This is ofgreat importance to accurately predict the initial conditionsof the string’s vibration. The model is found particularlyeffective to give results very close to experimental ones,especially when the player plays rapidly.

Further work will be carried out to investigate harpsi-chord voicing. Several plectra of various shapes will be

used for the measurements. A parametric study on theplectrum’s geometrical parameters will be helpful to pointout the relationship between the voicing process and theproduced sound as well as the instrumentalist control. Fur-ther, refinements of the current plectrum-string interactionmodel will be proposed. For instance, the friction forcesduring the slipping phase will be modeled in order to im-prove the prediction of the initial conditions of the stringfree oscillations. Besides, as the plectrum-string interac-tion is obviously dependent on the musical context furthercomparative analyses will be performed on various playingtechniques and music styles.

Acknowledgments

The authors acknowledge the Ministere de la culture et dela communication for its financial support to the project, aswell as Jean-Claude Battault and Stephane Vaiedelich forthe fruitful discussions, Jean-Marc Fontaine for taking thepictures of the plectrum, Laurent Quartier for his technicalhelp and Hugo Scurto for the preliminary tests.

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